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\title[ch05]{Chapter 11: CHARACTERISTIC VARIETIES}
\author[]{SCC}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

\begin{document}

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% 封面页
\begin{frame}
  \titlepage
\end{frame}

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% 目录页
\begin{frame}{Contents}
  \tableofcontents
\end{frame}

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% Section 1
\section{THE CHARACTERISTIC VARIETY.}
%---------------------------------------------------
\begin{frame}{1.1 LEMMA. }

Let $A_n$ be the $n$-th complex Weyl algebra and 
let $M$ be a finitely generated left $A_n$-module with a good filtration $\Gamma$. 
Then $\text{gr}^{\Gamma}M$ is a finitely generated module over the polynomial ring $S_n$. 
Let $\text{ann}(M,\Gamma)$ stand for the annihilator of $\text{gr}^{\Gamma}M$ in $S_n$. 
Then $\text{ann}(M,\Gamma)$ is an ideal of $S_n$. 
Let $\Omega$ be another good filtration of $M$. 
Then
$$
\text{rad}(\text{ann}(M,\Gamma)) = \text{rad}(\text{ann}(M,\Omega)).
$$

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{1.2 PROPOSITION. }

The ideal $\mathcal{I}(M) = \text{rad}(\text{ann}(M,\Gamma))$ is called the {\color{red}characteristic ideal} of $M$. 

The affine variety 
$$
\text{Ch}(M) = \mathcal{Z}(\mathcal{I}(M)) \subseteq \mathbb{C}^{2n}
$$
is called the {\color{red}characteristic variety} of $M$. 

Let $M$ be a finitely generated left $A_n$-module and $N$ a submodule of $M$. Then
$$
\text{Ch}(M) = \text{Ch}(N) \cup \text{Ch}(M/N).
$$

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{1.3 THEOREM. }

Let $M$ be a finitely generated left module over $A_n$. Then $\dim \text{Ch}(M) = d(M)$.

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% Section 2
\section{SYMPLECTIC GEOMETRY.}
%---------------------------------------------------
\begin{frame}{2.2 PROPOSITION. }

If $V$ is an affine variety of $\mathbb{C}^{2n}$, 
then we will say that it is {\color{red}involutive} 
if the tangent space $T_p(V) \subseteq \mathbb{C}^{2n}$ is {\color{red}involutive} (i.e., it contains its skew-complement)
at every non-singular point $p \in V$.
In particular, a hypersurface will always be involutive. 

This proposition says that the dimension of an involutive variety of $\mathbb{C}^{2n}$ is greater than or equal to $n$.

\noindent\rule{\textwidth}{0.4pt}


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%---------------------------------------------------
\begin{frame}{2.2.b DEFINITION. }

Let $\Phi: \mathbb{C}^{2n} \to (\mathbb{C}^{2n})^*$ be the linear map 
which associates $\omega(w,-)$ to $w \in \mathbb{C}^{2n}$.

Let $\mathrm{I}$ denote the inverse of the map $\Phi$. 

The {\color{red}Poisson bracket} of $f,g \in S_n$ is
$$
\{f,g\}(p) = \omega(\mathrm{I}d_pf,\mathrm{I}d_pg) = d_pf(\mathrm{I}d_pg).
$$

An explicit calculation using coordinates shows that
$$
\{f,g\}(p) = \sum_{1}^{n}\left\{\frac{\partial f}{\partial y_{n+i}}(p) \cdot \frac{\partial g}{\partial y_i}(p) - \frac{\partial g}{\partial y_{n+i}}(p) \cdot \frac{\partial f}{\partial y_i}(p)\right\}.
$$

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\end{frame}

%---------------------------------------------------
\begin{frame}{2.3 LEMMA. }

Let $V$ be an affine variety of $\mathbb{C}^{2n}$ and $p$ a point of $V$. 
If $\theta$ is a form on $\mathbb{C}^{2n}$ whose restriction to $T_p(V)$ is zero, 
then there exists $f \in \mathcal{I}(V)$ such that $\theta = d_pf$.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{2.4 PROPOSITION. }

An affine variety $V$ of $\mathbb{C}^{2n}$ is involutive if and only if 
its ideal $\mathcal{I}(V)$ is closed for the Poisson bracket.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{2.5 THEOREM. }

Let $M$ be a finitely generated left $A_n$-module. 
Then $\text{Ch}(M)$ is involutive with respect to the standard symplectic structure of $\mathbb{C}^{2n}$. 
Equivalently, $\mathcal{I}(M)$ is closed for the Poisson bracket.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{2.6 COROLLARY. }

Let $M$ be a finitely generated left $A_n$-module. Then
$$
d(M) = \dim \text{Ch}(M) \geq n.
$$

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\end{frame}

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% Section 3
\section{NON-HOLONOMIC IRREDUCIBLE MODULES.}
%---------------------------------------------------
\begin{frame}{3.0 STAFFORD'S EXAMPLE. }

If $n > 2$ and $\lambda_2,\ldots,\lambda_n \in \mathbb{C}$ are 
algebraically independent over $\mathbb{Q}$, then the operator
$$
s = \partial_1 + \left(\sum_{2}^{n} \lambda_i x_i \partial_i + x_i\right) + \sum_{2}^{n}(x_i - \partial_i)
$$
generates a maximal left ideal of $A_n$. 

%Let $M = A_n/A_ns$. Then $M$ is irreducible and $d(M) = 2n-1 > n$, since $n > 2$. 

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{3.1 PROPOSITION. }

Let $d$ be an operator of degree $k$ in $A_n$ and suppose that:
\begin{enumerate}
    \item the symbol $\sigma_k(d)$ is irreducible in $S_n$;
    \item the hypersurface $\mathcal{Z}(\sigma_k(d))$ is {\color{red}minimal} 
    (i.e., it does not contain a proper involutive homogeneous subvariety).
\end{enumerate}
Then the left ideal $A_nd$ is maximal. 
In particular, the quotient $A_n/A_nd$ is an irreducible module of dimension $2n-1$ over $A_n$.

\noindent\rule{\textwidth}{0.4pt}

Let $d=\partial_x^2+\partial_y^2+\partial_z^2$. 
The degree $k=2$. 
The symbol $\sigma_2(d)=\xi^2+\zeta^2+\eta^2$.
$S_3=k[x,y,z,\xi,\zeta,\eta]$. 


\end{frame}

%---------------------------------------------------
\begin{frame}{3.2 THEOREM. }

Let $S_n(k)$ be the homogeneous component of degree $k$ of $S_n$. 
This is a finite dimensional complex vector space; 
so it makes sense to talk about hypersurfaces in $S_n(k)$. 

We say that a property $\mathbf{P}$ holds {\color{red}generically} in $S_n(k)$ if the set
$$
\{f \in S_n(k) : \mathbf{P} \text{ does not hold for } f\}
$$
is contained in the union of countably many hypersurfaces of $S_n(k)$.

This theorem says that the property '$\mathcal{Z}(f)$ is minimal' holds generically in $S_n(k)$, 
whenever $k \geq 4$ and $n \geq 2$.

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% Section 4
\section{EXERCISES.}
%---------------------------------------------------
\begin{frame}{EXERCISE 1. }

Let $M = A_1/A_1x$. Let $\Gamma$ be the filtration of $M$ induced by $\mathcal{B}$ 
and $\Omega$ the filtration defined by $\Omega_k = B_k \cdot (\theta + A_1x)$. 
Show that $ann(M,\Gamma) = S_1y_1$ and $ann(M,\Omega) = S_1y_1^2 + S_1y_1y_2$. 
Compute their radicals.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 2. }

Show that if $\mathcal{J}$ is a homogeneous ideal of a graded algebra $R$ 
then $\text{rad}(\mathcal{J})$ is also homogeneous.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 3. }

Let $\mathcal{J}$ be a left ideal of $A_n$ and put $M = A_n/\mathcal{J}$. 
Show that $\text{Ch}(M) = \mathcal{Z}(\text{gr}(\mathcal{J}))$.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 4. }

Is the union of two involutive varieties involutive? What about their intersection?

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 5. }

Show that if $V$ is an involutive homogeneous variety of $\mathbb{C}^{2n}$ 
then its irreducible components are also homogeneous and involutive.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 6. }

Let $s$ be the operator of $A_n$ defined in §3. Let $M = A_n/A_ns$.
\begin{enumerate}
\item Compute $\text{Ch}(M)$.
\item Is it an irreducible variety of $\mathbb{C}^{2n}$?
\item Is it a minimal hypersurface?
\end{enumerate}

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 7. }

Let $\mathcal{J}$ be a left ideal of $A_n$. 
Show that if $\text{gr}(\mathcal{J})$ is a prime ideal of $S_n$ and $\mathcal{Z}(\text{gr}(\mathcal{J}))$ is minimal, 
then $\mathcal{J}$ is a maximal ideal of $A_n$.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 8. }

When is a hypersurface of $\mathbb{C}^{2n}$ lagrangian? 
Give an example of a lagrangian variety of $\mathbb{C}^{2n}$, when $n \geq 2$.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 9. }

A holonomic left $A_n$-module $M$ is {\color{red}regular} 
if there exists a filtration $\Gamma$ for $M$ 
such that $\text{ann}_{S_n} \text{gr}^\Gamma M$ is a {\color{red}radical ideal} of $S_n$. 
Let $N$ be a submodule of a regular holonomic module $M$. 
Show that $N$ and $M/N$ are also regular.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 10. }

Show that if $M$ is a {\color{red}regular holonomic} module whose characteristic variety is irreducible, 
then $M$ is an irreducible module. 
Why is the {\color{red}regular} hypothesis required?

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

%---------------------------------------------------
\begin{frame}{EXERCISE 11. }

Let $\mathcal{J}$ be an ideal of $S_n$. 
Show that $\mathcal{J}^2$ is always closed for the Poisson bracket.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}

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\end{document}
